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5 votes

$C(K_1)\cong C(K_2)$ if and only if $K_1\cong K_2$

This is the Banach-Stone theorem. You can find a proof in Conway's book (Theorem 2.1. page 172) The idea is that if $T \colon C(X) \to C(Y)$ is an onto isometry then $T^* \colon M(Y) \to M(X)$ is a $w^...
Evangelopoulos Foivos's user avatar
3 votes

Show that $\sum_{k=1}^n{2^{2k-1}\binom{2n+1}{2k}B_{2k}(0)}=n$

I'm giving credits to @darijgrinberg because I was using the Bernoulli polynomials to approach the proof and I finally figured it out thanks to his comment. Bernoulli polynomials are defined through ...
Conreu's user avatar
  • 2,688
3 votes

Good rigorous reference for wave phenomena

"Linear and Nonlinear Waves" by G. B. Whitham (Wiley) fits the bill here. It's intended for first and second year graduate students, but it's approachable enough that someone who is familiar ...
Alex Jones's user avatar
  • 9,428
2 votes

References and useful results on continuous one-parameter intersection of algebraic surfaces

It doesn't matter too much whether you take $\gamma$ to be a curve or something more arbitrary, due to the existence of space-filling curves and so forth. The fact that the $P_t$ are a parametric ...
Qiaochu Yuan's user avatar
2 votes

Nonstandard Analysis research project ideas

There are two approaches to non-standard analysis: (1) the "extension" approach and (2) the axiomatic approach. In the "extension" approach, the real numbers are extended to the ...
Mikhail Katz's user avatar
  • 43.8k
2 votes

Looking for an applied mathematics book with each chapter with an epigraph from The Simpsons

Googling "ODE System Editor" "Figure 6.9" gives one hit, to . This appears to be a chapter of the book in question (which is not ...
Michael Lugo's user avatar
2 votes

Is this proof of the angle bisector theorem known?

This seems a good approach. Euclid proves the angle bisector theorem in Elements VI, 3 before developing the theory of similar triangles. But his theory of similar triangles does not require Prop. VI,...
Edward Porcella's user avatar
2 votes

Contour Integral solution to differential equations, Euler transformation?

I know this method under the umbrella term of "integral representations of solutions to linear ODEs" or "Laplace's method". You're right, you can use a variety of kernels with the ...
brady's user avatar
  • 66
2 votes

Laplacian matrix for a multi graph

Your definition: $$L_{ij} = \begin{cases}-m_{ij} &\text{ if } i\neq j\\ \text{d}_i &\text{ else } \end{cases}$$ where $m_{ij}$ is the multiplicity of the edges between $i$ and $j$, is the most ...
caduk's user avatar
  • 4,870
1 vote

Reference Needed: Existence of Subsets of $\mathbb{N}$ with Specified Lower and Upper Asymptotic Densities

If $n_k$ is a sufficiently rapidly increasing sequence and you alternate between choosing $\alpha$-dense and $\beta$-dense subsets in $[n_k, n_{k+1}]\cap\mathbb{N}$, your resulting set will have these ...
TheAlertGerbil's user avatar
1 vote

Seeking "900 Geometry Problems" Book – Any Leads on Its Whereabouts?

I think you heard it a little wrong. The title is "110 Geometry Problems for the International Mathematical Olympiad" authored by Titu Andreescu & Cosmin Pohoata Check out : https://www....
Prem's user avatar
  • 12.3k
1 vote

Show that $\sum_{k=1}^n{2^{2k-1}\binom{2n+1}{2k}B_{2k}(0)}=n$

We know that $$ \frac{t}{e^t-1} = \sum\limits_{k=0}^\infty B_k \frac{t^k}{k!} $$ We need to find $$ \sum\limits_{k=0}^n 2^{2k-1} \binom{2n+1}{2k} B_{2k} = \frac{(2n+1)!}{2} \sum\limits_{k=0}^n \frac{2^...
Oleksandr  Kulkov's user avatar
1 vote

Proof of the General Change of Variables Theorem in Rn?

After a long search I found the demonstration of this theorem, with stronger hypotheses, in the following reference: [1] Bojarski, Bogdan, and Tadeusz Iwaniec. "Analytical foundations of the ...
Patrick Oliveira's user avatar
1 vote

Finding the Max of a Discrete Probability Distribution

So here is a general sketch of how you can solve the problem. Consider the simpler case that $N=2$, so that there are only two possible values $X_1$ and $X_2$, where $P(X_1) > P(X_2).$ Then if you ...
Alan Chung's user avatar
  • 1,232

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